Problem: $ C = \left[\begin{array}{rr}-1 & 1 \\ 2 & 1\end{array}\right]$ $ E = \left[\begin{array}{rrr}4 & 2 & 1 \\ 0 & -2 & 3\end{array}\right]$ What is $ C E$ ?
Answer: Because $ C$ has dimensions $(2\times2)$ and $ E$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ C E = \left[\begin{array}{rr}{-1} & {1} \\ {2} & {1}\end{array}\right] \left[\begin{array}{rrr}{4} & \color{#DF0030}{2} & \color{#9D38BD}{1} \\ {0} & \color{#DF0030}{-2} & \color{#9D38BD}{3}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{-1}\cdot{4}+{1}\cdot{0} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{-1}\cdot{4}+{1}\cdot{0} & ? & ? \\ {2}\cdot{4}+{1}\cdot{0} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{-1}\cdot{4}+{1}\cdot{0} & {-1}\cdot\color{#DF0030}{2}+{1}\cdot\color{#DF0030}{-2} & ? \\ {2}\cdot{4}+{1}\cdot{0} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{-1}\cdot{4}+{1}\cdot{0} & {-1}\cdot\color{#DF0030}{2}+{1}\cdot\color{#DF0030}{-2} & {-1}\cdot\color{#9D38BD}{1}+{1}\cdot\color{#9D38BD}{3} \\ {2}\cdot{4}+{1}\cdot{0} & {2}\cdot\color{#DF0030}{2}+{1}\cdot\color{#DF0030}{-2} & {2}\cdot\color{#9D38BD}{1}+{1}\cdot\color{#9D38BD}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-4 & -4 & 2 \\ 8 & 2 & 5\end{array}\right] $